Quality Alr Conditioning manufactures three home air conditioners: an economy model, a standard model, and a deluxe model. The profits per unit are $$63,$$$95,$ and $$135,$ respectively. The production requirements per unit are as follows:

$\backslash$table$[[,\backslash$table$[[$Number of$],[$Fans$]],\backslash$table$[[$Number of$],[$Cooling Coils$]],\backslash$table$[[$Manufacturing$],[$Time $($hours$)]]],[$Economy$,1,1,8],[$Standard$,1,2,12],[$Deluxe$,1,4,14]]$

For the coming production period, the company has $220$ fan motors, $340$ cooling coils, and $2,600$ hours of manufacturing time available. How many economy models $(E),$ standard models $($S$),$ and deluxe models $(D)$ should the company produce in order to maximize profit? The linear programming model for the problem is as follows:

Max $63E+95S+135D$

$s.t.$

$1E+1S+1D220$ Fan motors

$1E+2S+4D340$ Cooling coils

$8E+12S+14D2,600$ Manufacturing time

The computer solution is shown below.

$])$:$\}$

$E100\mathrm{.}00000,0\mathrm{.}00000$

$S120\mathrm{.}00000,0\mathrm{.}00000$

$D0\mathrm{.}00000,-24\mathrm{.}00000$

Constraint Slac$\frac{k}{S}$urplus Dual Value

$1,0\mathrm{.}00000,31\mathrm{.}00000$

$2,0\mathrm{.}00000,32\mathrm{.}00000$

$[3,360\mathrm{.}00000,0\mathrm{.}00000$

$\backslash$table$[[$Variable$,\backslash$table$[[$Objective$],[$Coefficient$]],\backslash$table$[[$Allowable$],[$Increase$]],\backslash$table$[[$Allowable$],[$Decrease$]]],[$E$,63\mathrm{.}00000,12\mathrm{.}00000,15\mathrm{.}50000],[$S$,95\mathrm{.}00000,31\mathrm{.}00000,8\mathrm{.}00000],[$D$,135\mathrm{.}00000,24\mathrm{.}00000,$Infinite$],[,,,],[$Constraint$,$RHS$,$Allowable,Allowable$],[1,$Value,Increase,Decrease$],[2,220\mathrm{.}00000,90\mathrm{.}00000,50\mathrm{.}00000],[3,340\mathrm{.}00000,90\mathrm{.}00000,120\mathrm{.}00000],[,2600\mathrm{.}00000,$Infinite,$360\mathrm{.}00000]]$

$($a$)$ Identify the range of optimality for each objective function coefficient. $($If there is no upper or lower limit$,$ enter NO LIMIT.$)$

$\backslash$table$[[E,$to$],[S,$to$],[D,$to$]]$

$($b$)$ Suppose the profit for the economy model is increased by $$6$ per unit, the profit for the standard model is decreased by $$2$ per unit, and the profit for the deluxe model is increased by $$4$ per unit. What will the new optimal solution be$?$

E units

$S$ units

D $|$units

profit

$($c$)$ Identify the range of feasibility for the right$-$hand$-$side values. $($If there is no upper or lower limit$,$ enter NO LIMIT.$)$

constraint $1$ to

constraint $2$

to

constraint $3$ to

$($d$)$ If the number of fan motors available for production is increased by $110,$ will the dual value for that constraint change? Explain.

Yes, the dual value will change because $110$ is greater than the allowable increase of $12.$

Yes, the dual value will change because $110$ is greater than the allowable increase of $90.$